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Critical random walks on two-dimensional complexes with applications to polling systems

MacPhee, I.M.; Menshikov, M.V.

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Authors

I.M. MacPhee



Abstract

We consider a time-homogeneous random walk Xi = {xi (t)} on a two-dimensional complex. All of our results here are formulated in a constructive way. By this we mean that for any given random walk we can, with an expression using only the first and second moments of the jumps and the return probabilities for some transient one-dimensional random walks, conclude whether the process is ergodic, null-recurrent or transient. Further we can determine when pth moments of passage times tau(K) to sets S-K = {x: parallel toxparallel to less than or equal to K} are finite (p > 0, real). Our main interest is in a new critical case where we will show the long-term behavior of the random walk is very similar to that found for walks with zero mean drift inside the quadrants. Recently a partial case of a polling system model in the critical regime was investigated by Menshikov and Zuyev who give explicit results in terms of the parameters of the queueing model. This model and some others can be interpreted as random walks on two-dimensional complexes.

Citation

MacPhee, I., & Menshikov, M. (2003). Critical random walks on two-dimensional complexes with applications to polling systems. Annals of Applied Probability, 13(4), 1399-1422. https://doi.org/10.1214/aoap/1069786503

Journal Article Type Article
Publication Date Nov 1, 2003
Deposit Date Apr 23, 2007
Publicly Available Date Mar 28, 2024
Journal Annals of Applied Probability
Print ISSN 1050-5164
Publisher Institute of Mathematical Statistics
Peer Reviewed Peer Reviewed
Volume 13
Issue 4
Pages 1399-1422
DOI https://doi.org/10.1214/aoap/1069786503
Keywords Random walk, Two-dimensional complex, Transience, Recurrence, Passage time moments, Polling systems.

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